Spacetime and Geometry

An Introduction to General Relativity

Spacetime and Geometry is a graduate-level textbook on general relativity, published by Addison-Wesley.

“For if each Star is little more a mathematical Point,
located upon the Hemisphere of Heaven by Right Ascension and Declination,
then all the Stars, taken together, tho’ innumerable,
must like any other set of points,
in turn represent some single gigantick Equation,
to the mind of God as straightforward as, say, the Equation of a Sphere,—
to us unreadable, incalculable.
A lonely, uncompensated, perhaps even impossible Task,—
yet some of us must ever be seeking, I suppose.”— Thomas Pynchon, Mason & Dixon

About the Book

In 1996 I taught a one-semester graduate course graduate course in general relativity at MIT. Along the way I typed up a detailed set of lecture notes. The book Spacetime and Geometry is a significantly revised and expanded version of these notes; about half of the finished book is completely new. The lecture notes will continue to be available for free online.

The philosophy of the book is to provide an accessible, useful, and pedagogical introduction to general relativity. In particular, no effort has been made to write a comprehensive reference book. More on the approach taken can be found in the preface.

Preface

General relativity is the most beautiful physical theory ever invented. It describes one of the most pervasive features of the world we experience — gravitation — in terms of an elegant mathematical structure — the differential geometry of curved spacetime — leading to unambigous predictions that have received spectacular experimental confirmation. Consequences of general relativity, from the big bang to black holes, are often what get young people first interested in physics, and it is an unalloyed joy to finally reach the point in one’s studies where these phenomena may be understood at a rigorous quantitative level. If you are contemplating reading this book, that point is here.

In recent decades, general relativity (GR) has become an integral and indispensable part of modern physics. For a long time after it was proposed by Einstein in 1916, GR was counted as a shining achievement that lay somewhat outside the mainstream of interesting research. Increasingly, however, contemporary students in a variety of specialties are finding it necessary to study Einstein’s theory. In addition to being an active research area in its own right, GR is part of the standard syllabus for anyone interested in astrophysics, cosmology, string theory, and even particle physics. This is not to slight the more pragmatic uses of GR, including the workings of the Global Positioning System (GPS) satellite network.

There is no shortage of books on GR, and many of them are excellent. Indeed, approximately thirty years ago witnessed the appearance of no fewer than three books in the subject, each of which has become a classic in its own right: those by Weinberg (1972), Misner, Thorne, and Wheeler (1973), and Hawking and Ellis (1975). Each of these books is suffused with a strongly-held point of view advocated by the authors. This has led to a love-hate relationship between these works and their readers; in each case, it takes little effort to find students who will declare them to be the best textbook ever written, or other students who find them completely unpalatable. For the individuals in question, these judgments may very well be correct; there are many different ways to approach this subject.

The present book has a single purpose: to provide a clear introduction to general relativity, suitable for graduate students or advanced undergraduates. I have attempted to include enough material so that almost any one-semester introductory course on GR can find the appropriate subjects covered in the text, but not too much more than that. In particular, I have tried to resist the temptation to write a comprehensive reference book. The only goal of this book is to teach you GR.

An intentional effort has been made to prefer the conventional over the ideosyncratic. If I can be accused of any particular ideological bias, it would be a tendency think of general relativity as a field theory, a point of view which helps one to appreciate the connections between GR, particle physics, and string theory. At the same time, there are a number of exciting astrophysical applications of GR (black holes, gravitational lensing, the production and detection of gravitational waves, the early universe, the late universe, the cosmological constant), and I have endeavored to include at least enough background discussion of these issues to prepare students to tackle the current literature.

The primary question facing any introductory treatment of general relativity is the level of mathematical rigor at which to operate. There is no uniquely proper solution, as different students will respond with different levels of understanding and enthusiasm to different approaches. Recognizing this, I have tried to provide something for everyone. I have not shied away from detailed formalism, but have also attempted to include concrete examples and informal discussion of the concepts under consideration. Much of the most mathematical material has been relegated to appendices. Some of the material in the appendices is actually an integral part of the course (for example, the discussion of conformal diagrams), but an individual reader or instructor can decide just when it is appropriate to delve into them; signposts are included in the body of the text.

Surprisingly, there are very few formal prerequisites for learning general relativity; most of the material is developed as you go along. Certainly no prior exposure to Riemannian geometry is assumed, nor would it necessarily be helpful. It would be nice to have already studied some special relativity; although a discussion is included in Chapter One, its purpose is more to review the basics and and introduce some notation, rather than to provide a self-contained introduction. Beyond that, some exposure to electromagnetism, Lagrangian mechanics, and linear algebra might be useful, but the essentials are included here.

The structure of the book should be clear. The first chapter is a review of special relativity and basic tensor algebra, including a brief discussion of classical field theory. The next two chapters introduce manifolds and curvature in some detail; some motivational physics is included, but building a mathematical framework is the primary goal. General relativity proper is introduced in Chapter Four, along with some discussion of alternative theories. The next four chapters discuss the three major applications of GR: black holes (two chapters), perturbation theory and gravitational waves, and cosmology. Each of these subjects has witnessed an explosion of research in recent years, so the discussions here will be necessarily introductory, but I have tried to emphasize issues of relevance to current work. These three applications can be covered in any order, although there are interdependencies highlighted in the text. Discussions of experimental tests are sprinkled through these chapters. Chapter Nine is a brief introduction to quantum field theory in curved spacetime; this is not a necessary part of a first look at GR, but has become increasingly important to work in quantum gravity and cosmology, and therefore deserves some mention. On the other hand, a few topics are scandalously neglected; the initial value problem and cosmological perturbation theory come to mind, but there are others. Fortunately there is no shortage of other resources. The appendices serve various purposes: there are discussions of technical points which were avoided in the body of the book, crucial concepts which could have been put in various different places, and extra topics which are useful but outside the main development.

Since the goal of the book is pedagogy rather than originality, I have often leaned heavily on other books (listed in the bibliography) when their expositions seemed perfectly sensible to me. When this leaning was especially heavy, I have indicated it in the text itself. It will be clear that a primary resource was the book by Wald (1984), which has become a standard reference in the field; readers of this book will hopefully be well-prepared to jump into the more advanced sections of Wald’s book.

This book grew out of a set of lecture notes that were prepared when I taught a course on GR at MIT. These notes are available on the web for free, and will continue to be so; they will be linked to the website listed below. Perhaps a little over half of the material here is contained in the notes, although the advantages of owning the book (several copies, even) should go without saying.

Countless people have contributed greatly both to my own understanding of general relativity and to this book in particular — too many to acknowledge with any hope of completeness. Some people, however, deserve special mention. Ted Pyne learned the subject along with me, taught me a great deal, and collaborated with me the first time we taught a GR course, as a seminar in the astronomy department at Harvard; parts of this book are based on our mutual notes. Nick Warner taught the course at MIT from which I first learned GR, and his lectures were certainly a very heavy influence on what appears here. Neil Cornish was kind enough to provide a wealth of exercises, many of which have been included at the end of each chapter. And among the many people who have read parts of the manuscript and offered suggestions, Sanaz Arkani-Hamed was kind enough to go through the entire thing in great detail.

I would also like to thank everyone who either commented in person or by email on different parts of the book; these include Tigran Aivazian, Teodora Beloreshka, Ed Bertschinger, Patrick Brady, Peter Brown, Jennifer Chen, Michele Ferraz Figueiro, Eanna Flanagan, Jacques Fric, Ygor Geurts, Marco Godina, Monica Guica, Jim Hartle, Tamas Hauer, Mark Hoffman, Daniel Holz, Ted Jacobson, Akash Kansagra, Chuck Keeton, Arthur Kosowsky, Eugene Lim, Jorma Louko, Robert A. McNees, Hayri Mutluay, Simon Ross, Itai Seggev, Robert Wald, and Barton Zwiebach. Apologies are due to anyone I may have neglected to mention. And along the way I was fortunate to be the recipient of wisdom and perspective from numerous people, including Shadi Bartsch, George Field, Deryn Fogg, Ilana Harrus, Gretchen Helfrich, Mari Ruti, Maria Spiropulu, Mark Trodden, and of course my family. (This wisdom often came in the form, “What were you thinking?”) Finally, I would like to thank the students in my GR classes, on whom the strategies deployed here were first tested, and express my gratitude to my students and collaborators, for excusing my book-related absences when I should have been doing research.

My friends who have written textbooks themselves tell me that the first printing of a book will sometimes contain mistakes. In the unlikely event that this happens here, there will be a list of errata kept at the website for the book: http://spacetimeandgeometry.net/. The website will also contain other relevant links of interest to readers.

During the time I was working on this book, I was supported by the National Science Foundation, the Department of Energy, the Alfred P. Sloan Foundation, and the David and Lucile Packard Foundation.

About 50% of the book is completely new; I’ve also polished and improved many of the explanations, and made the organization more flexible and user-friendly. The notes as they are will always be here for free.[/one_half_first]

[one_half][/one_half]

These lecture notes are a lightly edited version of the ones I handed out while teaching Physics 8.962, the graduate course in General Relativity at MIT, during Spring 1996. Each of the chapters is available here as PDF. Constructive comments and general flattery may be sent to me via the address below. The notes as a whole are available as gr-qc/9712019, and in html from “Level 5” at Caltech. What is even more amazing, the notes have been translated into French by Jacques Fric. Je ne parle pas francais, mais cette traduction devrait etre bonne.

Dates refer to the last nontrivial modification of the corresponding file (fixing typos doesn’t count). Note that, unlike the book, no real effort has been made to fix errata in these notes, so be sure to check your equations.

In a hurry? Can’t be bothered to slog through lovingly detailed descriptions of subtle features of curved spacetime? Try the No-Nonsense Introduction to General Relativity, a 24-page condensation of the full-blown lecture notes (PDF).

While you are here check out the Spacetime and Geometry page — including the annotated bibilography of technical and popular books, many available for purchase online.

Commentary/Reviews

The January 2005 issue of Physics Today has a joint review of my book and Jim Hartle’s book, by Jennie Traschen. It’s only available online to subscribers, unfortunately. An excerpt:

Carroll’s text adds a new option to the list of solid general relativity texts appropriate for graduate students planning to do research in gravity, high-energy theory, or both. … Of course, every teacher has a different opinion about the amount of foundational mathematics to include in a general relativity class that has to fit into one semester. Carroll has successfully steered a middle course, and I believe his text will be very useful to students. … Both books are fine contributions that extend in useful and different ways the range of pedagogical choices available to instructors.

My favourite part of the book is the first three chapters on differential geometry. The presentation of the mathematical formalism is crystal clear and very enjoyable, and it comes with a large number of helpful (and attractive) diagrams… The ninth and final chapter of Carroll’s book is devoted to a topic that has never been covered in an introductory text: quantum field theory in curved spacetime. To include this was a truly inspired thought, and Carroll is to be congratulated for this initiative… A complete education in general relativity cannot exclude this important subject, and now we have a textbook that presents it in a clear, accessible way.

Blurbs

Sean Carroll won the Graduate Student Council Teaching Award for the general relativity course that he taught at MIT, and the course has remained legendary among our graduate students. This book can be expected to spread the legend worldwide. Starting at the very beginning, Carroll gives a clear, well-motivated, and intellectually lively development of general relativity, extending into cosmology and even the basics of quantum field theory in curved spacetime. The book is a gem. –Alan Guth, Massachusetts Institute of Technology

Sean Carroll brings a relaxed authority to the teaching of general relativity at the graduate level. The text is beautifully written and succeeds in linking modern mathematical formalism to physical applications. Spacetime and Geometry continues the tradition of excellent relativity books and is likely to establish a common culture for the next generation of relativists. –Roger Blandford, Kavli Institute for Particle Astrophysics and Cosmology

This is an excellent, up to date, graduate text in general relativity, written in an informal yet sophisticated style. It provides an efficient introduction to the gravitational physics essential for today’s understanding of black holes, gravitational waves, and the universe. –James Hartle, University of California, Santa Barbara

This is the best book on contemporary gravitational physics for everyone from beginning graduate students to established researchers. Anyone interested in modern high-energy physics needs to understand cosmology and black holes, and this is the book they should read. It will be the leading textbook on gravity for years to come. –Renata Kallosh, Stanford University

“Spacetime and Geometry” is well-presented, very accessible and right up-to-date. It provides a comprehensive treatment that invites rather than discourages new students of the subject. –Bernard Schutz, Max Planck Institute for Gravitational Physics

A very clear, crisp, and self-contained introduction to general relativity in the modern language of differential geometry that’s a pleasure to read. Most remarkably, the author presents the most intricate subjects with characteristic clarity, but without compromising their integrity. If you have any doubts read and compare Carroll against any other text in the market. –Ben Grinstein, University of California, San Diego

Graduate General Relativity

Wald, General Relativity (Chicago, 1984).
Thorough discussions of a number of advanced topics, including black holes, global structure, and spinors. An invaluable reference, this is the book to turn to if you need the right answer to a well-posed GR question.

C. Misner, K. Thorne and J. Wheeler, Gravitation (Freeman, 1973).
The book that educated at least two generations of researchers in gravitational physics. Comprehensive and encyclopedic, the book is written in an often-ideosyncratic style that you will either like or not.

S. Weinberg, Gravitation and Cosmology (Wiley, 1972).
A great book at what it does, especially strong on astrophysics, cosmology, and experimental tests. However, it takes an unusual non-geometric approach to the material, and doesn’t discuss black holes. Weinberg is much better than most of us at cranking through impressive calculations.

J. Stewart, Advanced General Relativity (Cambridge, 2003).
A short but sweet introduction to some advanced topics, especially spinors, asymptotic structure, and the characteristic initial-value problem.

Mathematical Background

B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge, 1980).
Another good book by Schutz, this one covering some mathematical points that are left out of the GR book (but at a very accessible level). Included are discussions of Lie derivatives, differential forms, and applications to physics other than GR.

B.S. Ryden, An Introduction to Cosmology (Addison-Wesley, 2002).
A very modern and physical introduction to topics in contemporary cosmology, aimed at advanced undergraduates or beginning graduate students.

C.V. Johnson, D-Branes (Cambridge, 1998).
A detailed introduction to the extended objects called D-branes which have become an indispensable part of string theory; prior knowledge of string theory itself not required.

L. Smolin, Three Roads to Quantum Gravity (Perseus, 1998).
The “three roads” are string theory, loop quantum gravity, and something more profound; Smolin is a partisan for loop quantum gravity, but the discussion should be interesting for everyone.

Applications of Classical Physics, Roger Blandford and Kip Thorne
An illuminating overview of topics in classical physics by two masters. (Advanced undergraduate level; GR is covered briefly at the end.)

SPIRES High-Energy Physics Literature Database
A service which not only indexes papers, but keeps track of references and citations. Nominally devoted to high-energy physics, but gradually expanding to keep track of gravitation and astrophysics.

Errata

This page collects any mistakes that people have been able to find in the book. Those listed as being noticed before 10/14/03 have been corrected in the second printing. Dates refer to when the addition was made to this page, not necessarily when it was sent to me.

p. 49: In the middle of the page, “accelerating at a constant velocity” should be “accelerating at a constant rate,” of course. [Thanks to D. Hartmann, 11/28/03.]

p. 52: In the paragraph beginning “Consider”, each appearance of “box” should be “rocket”, to agree with Figure 2.2. [Thanks to D. Taylor, 4/3/04.]

p. 57: I goofed on the cone, when I said it was a non-smooth manifold. The metric on the cone, inherited from its embedding in Euclidean space, is not smooth at the vertex, but it is perfectly possible to give the cone a smooth atlas (just project it down to the plane, and use the conventional atlas there). The irony is that I knew all of this perfectly well, but had somehow convinced myself that there was a more natural atlas for the cone, one that was not smooth. Upon reflection, I’m not sure where that conviction came from. [Thanks to P. Shepard and N. Jeevanje, communicated by R. Bousso, 1/29/04.]

p. 74: Near the beginning of the first full paragraph, “to see a sketch a proof” should just be “to sketch a proof.” [Thanks to J. Collins, 11/21/03.]

p. 79: Near the end of the first paragraph, “a curve of zero length is chronal but not causal” is exactly backwards, and should be “a curve of zero length is causal but not chronal.” In the same sentence, the word “in” should be inserted between “not necessarily” and “its own.” [Thanks to D. Taylor, 12/6/03.]

p. 80: In figure 2.24, the line from D^+(Sigma) should extend into the interior of the past light cone. [Thanks to D. Taylor, 12/11/03.]

p. 84: In line 5 of the first paragraph, “the the” should just be “the”. [Thanks to D. Taylor, 12/11/03.]

pp. 86-87: Maxwell’s equations are referenced as (1.88) and (1.89), while they are actually (1.96) and (1.97). [Thanks to D. Hartmann and J. Olson, 10/14/03.]

p. 87: After (2.86), “phi” should be capitalized for consistency with Chapter One. [Thanks to D. Taylor, 1/10/04.]

p. 87: A phrase was mistakenly dropped from the sentence containing (2.87), which should read “The other one of Maxwell’s equations, (1.96), can be expressed as an equation between three-forms: …” [10/31/03.]

p. 92: Problem 9 doesn’t make sense in “Euclidean three-space”, it should be in ordinary four-dimensional Minkowski space. Only the ball in part (d) is in Euclidean three-space. [Thanks to D. Starr, 2/10/04.]

Chapter Three

p. 96: After (3.11), it’s not precisely correct to say that the transformation properties of the tilded connection coefficients are the same as those of the original Gammas; in the analogue of (3.10), there would be a plus sign instead of a minus sign. [Thanks to D. Starr, 4/3/04.]

p. 99: In (3.20), the index on V in the second and third terms should be a nu, not a lambda. [Thanks to G. Chinnici, 4/3/04.]

p. 102: In (3.36), the second and third terms should be multiplied by a factor of 2, given the definition of the exterior derivative. [Thanks to D. Taylor, 1/10/04.]

p. 112: In the fifth-to-last line of the (only) full paragraph, the “mu” in “the sought-after coordinates x^mu” should have a hat. [Thanks to D. Taylor, 1/10/04.]

p. 108: The sentence containing (3.56) was mangled. After the semicolon should read “in fact, looking back to the expression (1.106) for the Lorentz force in special relativity, it is natural to guess that”, and then the equation. [Thanks to D. Hartmann, 11/28/03.]

p. 111: In the first sentence of the first full paragraph, “convenient a way” should just be “convenient way”. [Thanks to D. Hartmann, 11/28/03.]

p. 122: In the second line after (3.111), “the last term is simply the torsion tensor” would be better rendered as “the antisymmetrized connection coefficients in the last term are simply one-half times the torsion tensor”. [Thanks to D. Taylor, 2/11/04.]

p. 126: To be perfectly clear, at the end of the first paragraph of section 3.7 we should say “reduce the number of independent components” rather than just “reduce the independent components”. [Thanks to R. Riklund, 1/7/04.]

p. 130: After (3.148), instead of “the Appendices” we should have “Appendix G”. [Thanks to D. Taylor, 2/11/04.]

p. 136: The long sentence at the bottom of the page was apparently mangled in the typesetting process; delete the word “using” at the beginning of the sentence, and change the comma after “generalizes to additional indices” to a colon. Not my most elegant sentence in any event. [Thanks to D. Taylor, 2/11/04.]

p. 141: In (3.191), the nu in the last term should be an ordinary subscript, not a sub-subscript. [Thanks to D. Starr, 4/3/04.]

p. 145: At the end of the paragraph containing (3.204), the last sentence should include the condition “(when t is the proper time)”. [10/14/03.]

Chapter Four

p. 179: In the sixth line, “out” should be “our”. [Thanks to D. Taylor, 4/3/04.]

p. 190: In the very first line, “will suppressed” should be “will be suppressed”. [Thanks to T. Keech, 10/20/03.]

p. 190: In Exercise 1, just to be clear, let’s assume that the A_mu J^mu term does not contribute to the energy-momentum tensor. It won’t, for example, for charged point particles, since J^mu contains an implicit factor of 1/sqrt{-g}, as in the delta-function defined in (4.157). [Thanks to H. Haber, 4/5/04.]

p. 191: In Exercise 5, “the Appendices” could be more usefully rendered as “Appendices D and F”. [Thanks to D. Taylor, 4/3/04.]

pp. 221, 223: In Figures 5.10 and 5.11, one side of the middle light cones should be perfectly vertical, to line up with the event horizon at r=2GM. [Thanks to D. Taylor, 4/3/04.]

p. 225: “Kruskal-Szekres” should be “Kruskal-Szekeres”. [Thanks to R. Riklund, 1/7/04, and apologies to Prof. George Szekeres.]

Chapter Six

pp 255-273: Late in the game I decided to use rationalized (Heaviside-Lorentz) units for electromagnetism. In these units there are no 4pi’s in Maxwell’s equations (see p. 29) but there are in the expression for the electric field of a point charge (p. 457). The one place the change didn’t happen is in the discussion of charged (RN) black holes in Chapter Six. I believe that every appearance of the charges Q or P in that chapter should be divided by 4pi to make it okay. [Thanks to E. Abers, 4/3/04.]

Chapter Seven

p. 304: There should not be a minus sign in (7.134). [Thanks to E. Abers, 4/3/04.]

p. 375: Problem 2 was completely fumbled. First off, it should be “e^{2Ht}”, not “e^{Ht}” (if we want H to really be the Hubble parameter). Second and worse, comoving observers are precisely the wrong ones to ask this question about; for them, t actually is an affine parameter. The question should have asked about non-comoving observers. [Thanks to D. Garfinkle, 2/2/04.]

p. 384: In (9.34), that should be a time-ordered exponential, similar in spirit to the path-ordering discussed in Appendix I. [Thanks to R. Riklund, 4/3/04.]

p. 388: The left-hand-side of (9.57) really shouldn’t use mu as a dummy index twice. [Thanks to R. Riklund, 4/3/04.]

p. 390: There should not be a comma in the square root in (9.70). [Thanks to R. Riklund, 4/3/04.]

p. 403: The last term in (9.130) should be 1/alpha^2, rather than alpha^2. [Thanks to J. Fric, 11/26/03.]

Appendices

p. 425: The left arrow in Figure A.2 should be labeled “V(p)”. [Thanks to D. Starr, 4/3/04.]

p. 426: There should be parentheses around the last p in the caption to Figure A.3. [Thanks to D. Starr, 4/3/04.]

p. 432: In the caption of Figure B.3, at the end of the second line “T a point” should be “T at a point” . [Thanks to D. Starr, 4/3/04.]

p. 432: The pullbacks in (B.9) and (B.10) should have upper asterisks, not lower ones. [Thanks to G. Chinnici, 4/3/04.]

p. 436: In (B.24) and (B.25), the vector V should be thought of as infinitesimal; alternatively, think of V as finite and multiply each appearance by an infinitesimal parameter epsilon. [Thanks to D. Starr, 4/3/04.]

p. 436: Below (B.25), the definition of the energy-momentum tensor is actually (4.75), not (4.73). [Thanks to D. Starr, 4/3/04.]

pp. 437, 469: The Exercises in the Appendices are numbered incorrectly; the correct version should be obvious. [10/14/03.]

p. 439: In the definition of a submanifold, the condition on the inverse map should be that it is C^infinity, not that it is one-to-one (which it always will be). Also, it should have been made clear that this definition, following Wald’s GR book, differs from one common in the math literature; Wald was careful enough to point this out, while I was not. In much of the math literature, what I called an “immersed submanifold” would simply be an “immersion”; this makes sense, since if the image is self-intersecting it’s not really a manifold. What I called an “embedded submanifold” would only be an “immersed submanifold”, and the definition of “embedded” would include the additional condition that the image not come arbitrarily close to intersecting itself (technically, that the map phi defines a homeomorphism between S and its image). [Thanks to D. Starr, 4/3/04.]

p. 439: In third line of second paragraph, “the the” should be “the”. [Thanks to D. Snead, 12/23/03.]

p. 439: Each appearance of “phi[M]” in the second paragraph should be “phi[S]”. [Thanks to D. Snead and G. Chinnici, 12/23/03.]

p. 440: In the third line of the first full paragraph, “m function” should be “m functions”. [Thanks to D. Starr, 4/3/04.]

p. 441: In the first complete sentence after (C.4), “fit together to define a submanifold” should be “fit together to define a set of submanifolds”. [Thanks to D. Starr, 4/3/04.]

p. 443: In (D.2), the index on the derivative should be “nu”. [Thanks to D. Snead and I. Sahabandu, 12/21/03.]

p. 444: In the line between (D.10) and (D.11), the equation for zeta should be an equation for xi. [10/14/03.]

p. 451: Each appearance of “Codacci” should be “Codazzi.” [Thanks to M. Godina, 10/29/03.]